Is Arnold Schwarzenegger making in-roads in his rumored attempt to win back estranged wife Maria Shriver? On Tuesday, TMZ spotted the exes grabbing lunch at paparazzi-friendly eatery The Ivy at the Shore in Santa Monica. After the meal, they strolled over to a furniture store and browsed the home goods.

No word on whose digs they were shopping for, although Maria is reportedly renovating a $12 million Brentwood estate. She purchased the mansion after moving out of the family's nearby home following revelations last spring that Ahnuld had fathered a secret son more than 14 years ago with the family's longtime housekeeper, Mildred Baena. Shriver filed for divorce on July 1.

During their lunch outing, Arnold very much acted like the thoughtful companion. At one point, says the London Daily Mail, he fished out one of his "official" Governor's jackets from the trunk of his car when he saw that Maria was cold.

Gestures like this might explain why the New York Post believes they aren't yet ready to toss in the towel on their 25-year union, and claims they're now in couples therapy in a bid to work things out.

Sources contend that the Catholic-raised Shriver, 56, hasn't cottoned to the idea of divorce, while a repentant Ahnuld, 64, has realized "the error of his cheating ways."

"Arnold and Maria are having marriage counseling to work through their issues to get back together," says an insider. "Maria is a devout Catholic and doesn't believe in divorce, but is still struggling with the humiliation she endured by him."

And with their divorce proceedings moving at a snail's pace (the next court hearing is reportedly scheduled for 2016), Schwarzenegger has plenty of time to plead his case.

"He is promising the moon and telling Maria he has completely changed his ways," tattles another confidant. "Maria's friends are warning her to be careful, saying he'll never change."

Arnold, continues the snitch, "is desperate to win her back, mostly because he realizes he can't live without her, and partly because he doesn't have the political clout without her. Winning her back would be the best way to show the world he has been forgiven for his transgressions."

But it seems those transgressions may be insurmountable. A source maintains to Us Weekly that the pair, who have kept their split amicable for the sake of their four kids, are not in counseling or seeking to reunite.

"They will continue to be seen together because of their family," explains the spy, "but they are not reconciling."